Hermann Schubert Probes the Fourth Dimension

Mathematical Essays and Recreations

Hermann Schubert was a German teacher and textbook author. Mathematical Essays and Recreations is a small collection of articles ranging from the foundations of the number system, the foundations of algebra up to an extensive essay about the fourth dimension.

In the first article: Notion and definition of number, Shubert gives a brief account of how the concept of number evolved within the human mind.

In another of his essays: Monism in arithmetic, the author explores and writes about the elementary rules of algebra and about the importance that mathematical systems can operate with defined rules for any kind of number without making exceptions for some of them. For example, we can define the commutative rule of addition for the natural numbers, as in 2 + 3 = 3 + 2, and the same number should apply if instead, we use natural an imaginary numbers combined, like 2 + 3i = 3i + 2. Following his exposition, we can easily see why the class of complex numbers is such a robust field in mathematics, and how it can be derived from the natural numbers following easy and consistent steps. On the other hand, we can also see why the quaternions—when compared to the complex numbers—lack such popularity and why this class of number is so limited in applications and acceptance.

The third essay in this Datum edition is about the fourth dimension. But contrary to other science authors that rarely touch the fourth dimension from the metaphysical point of view, Schubert is not afraid to confront both doctrines. When dealing with the fourth dimension from the mathematical point of view, Schubert goes carefully from the definition of the point to the definition of dimension to a many-dimensioned space.

Are There Coordinates for the Fourth Dimension?

The interest in the fourth dimension is ever increasing. We all keep asking: Can there really exist a fourth dimension? In what direction should we look to find it? Why there are so many interpretations of it?

Is the spiritual fourth dimension the same as the physicists' interpretation? What is doing the mathematics about it?

The legs of a table do not have
any naming order. 

Well, to begin with, we all agree that WE ALL LIVE in three dimensions. That's a good start, but we do not all agree which one is THE FIRST DIMENSION. We don't know which dimension is the SECOND DIMENSION; so, how can we all agree which dimension should be the FOURTH DIMENSION?

To study the fourth dimension from the geometrical or algebraic standpoint of view we should associate dimensions with coordinates in a spatial hyperspace. This approach leads us to ask: Are there coordinates for the fourth dimension?

In the article, The Coordinates of the Fourth Dimension, I use the figure of a little dining table to ask which one of the four legs of the table we should say is THE FIRST LEG, which one of the four legs is the FOURTH LEG.

Take the challenge, read the article,  and be the first one to answer the question and fill in the blanks: In any table, the first leg is determined by ...!

Why Is It so Difficult to Reason About the Prime Numbers?

In the history of mathematics, the prime numbers are ancient as the invention of the natural numbers. To review a little, the natural numbers are the simple positive integers 1, 2, 3, ... —a series without end.

In this series of the natural numbers, when we are given some number, it is easy to tell what the next number is; just add 1 to the given number! (This explains why it is funny to ask elementary school kids for the biggest number he/she can think about, and then confront his answer by telling him/her to add 1 to the number he mentioned.)

On the other hand, the prime numbers are not so easy to visualize; their distribution is as if they were randomly distributed.

Statue of Euclid.

Let's review for a moment the standard definition of a prime number: a prime number is a natural number which has exactly two distinct natural number divisors: 1 and itself. For example, the primes less than 100 are:

    2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 31, 37, 41, 43, 47, 53, 59, 61, 67, 71, 73, 79, 83, 89, 97

The number 1 is by definition not a prime number. Note how unusual is the beginning of this series: the first prime is even --the only even number that is prime; all the other following primes are odd. But not every odd number is prime. To the number 3 follows 5; to 5 follows 7, but to 7 does not follow the number 9.

The simple task of finding the prime number that follows another prime impossible. With the prime numbers it happens that given any prime number, there is no way to compute the next prime that follows—or precede—the given one.

However, the primes are very special in one aspect: every natural number is either a prime or it is the product of some primes. This is a far-reaching assertion because it surreptitiously states that the natural numbers are in one way or another all made of prime numbers.

Euclid (ca. 300 BC), and Eratosthenes (ca. 200 BC) were the first two Greek mathematicians to work extensively with the properties of the prime numbers. Euclid proved that the primes were infinite (no way to find the last prime), and Eratosthenes devised a method to sieve out the primes from the series of the natural numbers.

Since then, the primes numbers are a constant headache for number theorists and mathematicians in general.

Historical Quotations About Prime Numbers

However, mathematics is not a private property of mathematicians and philosophers of science. What about the public? They also have something to say. What does the layman have to say about the prime numbers? Do politicians have something to add to this arduous field of mathematics (they always have something to say!)?

Let's see how some imaginary examples of how some famous people would have reasoned about if the number 9 is prime or not.

Christopher Columbus: “3 is prime, 5 is prime, 7 is prime. According to some ancient manuscripts, 9 is not a prime number, but beyond the distant horizon of the oceans, in the New World that I am going to discover, there are surely lots of them.”

Dimitri Mendeleev: “3 is a prime, 5 is a prime, and 7 is a prime, but 9 is a noble prime that deserves a separate row in the periodic table of the primes.”

Charlie Chaplin: “3 is a prime, 5 is a prime, 7 is a prime, 9 is the next prime after 8.”

John F. Kennedy: “1 is not a prime number and 9 is not a prime number? Then ask not what the primes can do for you, ask what you can do for the primes.”

Stephen Hawking: “2, 3, 5 and 7 are prime numbers: 9 is not prime, but in the black holes, past beyond the event horizon, anything can happen.”

George W. Bush: “3 is prime, 5 is prime, 7 is prime, and 9 … well, any odd number can be prime as long as it is not 9.”


For more examples about how easily is to be misled when reasoning about the series of the prime numbers, then see the article:

Historical Quotations About Prime Numbers

Transcomplex Numbers

Foundations of Transcomplex Numbers.

Integer numbers, negative numbers, fractions, real numbers, transcendental numbers, irrational numbers, and imaginary numbers are a few of the number types we usually find in mathematics.

Is there no end to this? Is there no "final" type of numbers?

From the standpoint of number fields, all of them can be encompassed into one type called the complex number field.

Wikipedia, in a short background, mentions how the complex numbers emerged:

Complex numbers were first conceived and defined by the Italian mathematician Gerolamo Cardano, who called them "fictitious", during his attempts to find solutions to cubic equations. The solution of a general cubic equation may require intermediate calculations containing the square roots of negative numbers, even when the final solutions are real numbers, a situation known as casus irreducibilis. This ultimately led to the fundamental theorem of algebra, which shows that with complex numbers, a solution exists to every polynomial equation of degree one or higher.

Complex numbers can also be understood and developed from the standpoint of view of ordered pairs. Today, developing the complex number system from the foundations of set theory and the concept of ordered pairs is possibly the most intuitive approach we have at hand.

For a rigorous development of the complex number system download the free EBook: Foundations Of Transcomplex Numbers: An Extension Of The Complex Number System To Four Dimensions.

This mathematics book is about a way of extending the complex numbers system to four-coordinate variables, maintaining the usual operations attributed to the complex numbers.

Foundations ... is a fully illustrated EBook. See --and Click-- for example, the following figure about how to multiply two ordered pairs:

Transcomplex numbers are an extension of
the common complex numbers. 

Complex numbers are usually plotted using the familiar plane Cartesian coordinate system, but transcomplex numbers are four-entry ordered pairs, also called 4-tuples, so they belong to a four-dimensioned space.

In a nutshell, transcomplex numbers are complex numbers whose elements are ordered pairs.

In the following simple illustration, also taken from Foundations ... we can see that out of a four-entry complex number system we can extract four 3-dimensional spaces like "ours".

The transcomplex numbers need a 4-dimensional
coordinate system to be represented.

The chapters of the book are divided as follows:

• Ordered Pairs. The whole theory of transcomplex functions is based on the ordered pair concept: from the two-dimension plane up to the four-dimension space.
• Complex Numbers. The complex numbers system is derived from the ordered pair's concept.
• Transcomplex Numbers. Here starts the extension of the complex numbers into ordered pairs of complex numbers, arriving at the concept of transcomplexs.
• The Coordinate System S4. This chapter is devoted to deriving a suitable coordinate system to plot transcomplex functions.
• Transcomplex Functions. Functions of complex variables evolve to make space for functions of four-entries ordered pairs.
• Transcomplex Surfaces. A radical and totally new perception of surfaces generated by complex variables.
• Theorem Proofs. This chapter collects all the proofs of the theorems stated along the book.

Do Random Numbers Really Exist?

One Million Random Digits.

While I was collecting information for this month's post, I was also looking for new material for a new free E-Book to compile for my readers. The idea of the so-called random numbers sprung into my mind, so I began to search for this topic.

I found an interesting book review titled: A Million Random Digits with 100,000 Normal Deviates. The book was originally published in 1955 by the RAND Corporation, so the "review" was a little late, but its OK; the author was "reviewing" one of the oldest books in his library. This book can be found and read at Google Book Search.

Doing a deeper search I also found another article and an E-Mail by Mr. Nathan Kennedy complaining to the RAND Corp about their stand that the One Million Random Digits table was of their property and that it cannot be redistributed on the Internet. By a great coincidence, my intention was the same as Mr. Kennedy's, and his intention of putting the table on the Internet was the same as mine. However, and in a great unselfish gesture, Mr. Kennedy generated his own table of random digits and proceeded to place it on the Internet for free as a text file on the same page.

Since he authorized the use of his million digits table, I reformatted the text file as a PDF file, designed a cover page, wrote a small introduction for it, and made an E-Book, to be distributed also for free.

I am not a statistician, so maybe I will never find a practical use for this kind of numerical table, however, random numbers are of interest for me, and probably for many others, for the degree of strangeness they bear.

Are there really random numbers, or there are random events?

Can we really speak of random "numbers"? Isn't it more appropriate to speak of random "events"? Can we make at least some arithmetic operations with them? Can we add two RNs and still say that the sum is also "random"? Can we multiply them to obtain -without a doubt- that the result is also random?

Note that the RNs are not obtained by any formula, or equation, or matrix, or any predefined mathematical operation; they are mainly obtained by algorithms fed by some "physical" events like atmospheric variations, radioactive decay, thermal processes, or the like. Hence, what we are doing is using unpredictable physical events, assign to each "event" a number, and say that this is an RN. Another nonphysical source of random sequences of digits (but sometimes questioned) is by selecting digits or portions of digits of the decimal expression of irrational or transcendental numbers.

But without physical events, can we still generate RNs? There are some rudimentary approaches, but they are mostly mere mathematical curiosities.

The interested reader can find more authoritative articles at Random.org where he/she can obtain instant (real-time) random numbers for the lottery, cards, passwords, etc.

Paradoxes of the Infinite

The "Gabriel's Horn" is an ideal object
that can be finite and infinite at the same time.

Can an "object" be finite and infinite at the same time? Contrary to what our intuition dictates, it seems that this duality can arise in mathematics.

We are used to thinking that the "infinite" is a well-defined concept like some of our everyday ideas of "here", "there", etc. But, if that were the case, we would not have so many paradoxes arising from this field of science that Gauss referred to as the "queen of sciences".

Of course, George Cantor did a great contribution fixing the traditional and loose understanding of the infinite, especially introducing a scale of infinitudes when he proved that there is more than one kind of infinite. Since then, many other mathematicians and philosophers had been kept busy untangling some paradoxes that the new scale of the infinite had brought.

Among them, Jorge Luis Borges surfaced and worked literally with great success the problem of random infinite series without a first and the last term.

To learn about Borges' unusual understanding of the infinite follow this series of posts beginning with the post Nobody understand the infinite so well as Borges.

To learn about the unusual paradox of how can an "object" can be finite and infinite at the same time follow this article Top myths about the infinite about Torricelli's trumpet also called Gabriel's horn.

The Book of Sand of Borges and the Continuum of Cantor

The previous article

In the previous article entitled Nobody understands the infinite better than Borges I began the presentation of the short story The Book of Sand of the authorship of Jorge Luis Borges. In the story, The Book of Sand is a book that Borges acquired from a seller of books and Bibles that has the particularity that each time you open the book in one page, the page number changes, and the next page number is also changed. The book never had a first and last defined pages, and as if that were not enough, the text and illustrations never appear twice in the same place or on the same page.

The article ended like this:

Imagine that you open that "devilish book" and by some unknown power you can write the sequence of the page numbering as it appears page by page. Now close the book and reopen it again and repeat the process again. You will "finally" obtain all possible orderings of the natural numbers. I cannot show it here, but is possible to prove that your "list" of all possible orderings of the natural numbers is not countable, not even infinitely countable. Thus opening an closing The Book of Sand is an act of delving into the Continuum, an action that possibly Cantor never though of.

In the present article, which we can consider as a continuation of the one above, we will delve into an informal mathematical proof to show that the Book of Sand of Borges is much more than just an infinite book: it is a Transfinite book.

The infinitudes of Cantor

George Cantor conducted an extensive and deep research on the categories of the infinitude; something that was new to the mathematicians of his time. Prior to him it was assumed that all infinitudes were equal.

If we start with the series of the natural numbers 1, 2, 3, ... we say they are infinite; in fact, this series is the infinite series for excellence for its simplicity. But the even numbers series 2, 4, 6, ... it is also infinite, even when they seem to represent the "half" of the natural numbers. Another infinite set of integers is the set of prime numbers, despite the fact that as they progress the "distance" between them is widening. These sets were begun to be called "countably infinite" not because they were exactly "countable" but because they can be related in a one-to-one (1-1) relationship with the natural numbers we use to count.

More surprising is the fact that it can be shown that the fractions are also infinitely countable. Unexpectedly, with the proper arrangement of the elements, all fractions, such as 1/2, 4 /5, 458/245, ... can be infinitely listed, or counted as the first fraction, the second fraction, the third fraction, and so on.

Findings like these led to Cantor to deepen into the concept of "infinity" as we use it daily. Under a lot of opposition and humiliation, Cantor managed to formalize and establish the Theory of Sets, and the arithmetic of the infinite as a strong and indispensable field in the science of mathematics.

One of his sensational findings was demonstrating that there are infinite sets that are higher than the infinite set of natural numbers. There is no way to establish a one-to-one (1-1) correspondence between the natural numbers with those infinities, therefore, we must invent new categories for certain sets. These sets were called by Cantor Transfinite; infinities way beyond the infinity of the natural numbers.

The simplest example of transfinite sets is the set of the real numbers. Real numbers comprise those we commonly call decimal numbers and the infinite decimals that we never can end writing because they are

• either "irrationals" as the square root of 2, commonly symbolized as √(2) = 1.414213562373...,
• or they are transcendental numbers like the π =3.141592653589...

The discovery was unforeseen because it was not expected that the number of irrational numbers was "so big" as to need to coin a new term and concept to study them. Note the reader that the contribution of the finite decimal numbers is almost null because the finite decimals can be expressed as fractions and we already mentioned that the fractions are "countable". For example, since 0.500... = 1/2, 0.333... = 1/3, etc. we can take away all those decimals out of the real line and still the remaining reals are transfinite.

This new set of numbers that can not be "counted" with the set of all natural numbers was the one to be known as The Continuum.

The properties of the Continuum are incredible: in the same way as the set of all even numbers is infinite, despite being a subset of the natural numbers, there are also ways to create subsets of real numbers that are equally transfinite as the set of all the real numbers itself. For example, the "small" line segment between the decimal 0.1 and the decimal 1.0 contains a transfinite number of real numbers. To this segment, later, we will give a good use with the Book of Sand of Borges.

The Book of Sand is a Transfinite book

After this brief digression of going deeper into what are the real numbers, in view of the fact that we will need them later, let us now return to the Book of Sand and make a "thought experiment" with it. Recall that the main peculiarity of this enigmatic and esoteric book was that the pages were randomly enumerated and that it lacked a fixed first and last page because there were pages popping out of the nothingness at the beginning and at the end of the book.

Now imagine that you open this "devilish book" —as the Bible seller told Borges— and that for some unknown power you are able to write the sequence of the page numbers as they appear page by page. Now close the book and open it again and repeat the same process continuously. If you repeat this without stopping you will "finally" get all the possible combinations of sequences of pages of The Book of Sand.

What relationship exists between all the possible sequences of The Book of Sand of Borges and The Continuum of Cantor? Well, let's see if it is possible to establish unambiguous correspondences between all combinations of pages of The Book of Sand and The Continuum of Cantor.

There are as many instances of "The Books of Sand"
as there are real numbers.

To show the existence of unique relationships between the two sets requires a demonstration in two separate parts:

1. That we can produce a function such that for every "book of sand" the function can assign an unequivocal decimal point in the Continuum. This is the relation A in the illustration.
2. That there is also another function such that for every decimal in the Continuum the function can assign an unequivocal “book of sand”. That is the relation B in the illustration.

Will it be possible to demonstrate the existence of such mappings for the relations A and B as shown in the diagram?

This will be seen in the next post.

Selected Puzzles from Henry E. Dudeney

Henry E. Dudeney (1857-1930) was an English logician and mathematician that specialized in creating and collecting puzzles.

Two men arguing about how much liquid
is in a closed barrel without opening it.

Amusements in Mathematics, published by 1917 is a great collection of geometrical, chessboard and magic square problems.

The free E-book that I am offering you now is titled: 44 Selected Puzzles and Pastimes from Henry E. Dudeney. The puzzles are a hand-picked selection of the most "attractive" puzzles in the sense that some of Dudeney's problems are verbal, others are about chessboards, dominoes, etc., but this selection is all about those with some graphical appeal.

Download the free ebook: 44 Selected Puzzles and Pastimes from Henry E. Dudeney for free.
Click here

As an example of the EBook content take The Barrel Puzzle, one of the Dudeney's ingenious puzzle included in the EBook.

The puzzle goes like this

The men in the illustration are disputing over the liquid contents of a barrel. What the particular liquid is it is impossible to say, for we are unable to look into the barrel; so we will call it water. One man says that the barrel is more than half full, while the other insists that it is not half full. What is their easiest way of settling the point? It is not necessary to use stick, string, or implement of any kind for measuring. I give this merely as one of the simplest possible examples of the value of ordinary sagacity in the solving of puzzles. What are apparently very difficult problems may frequently be solved in a similarly easy manner if we only use a little common sense.

Clearly, this is a situation we may encounter some day. He clearly says that the content of the barrel does not matter: it can be oil, petroleum, wine, or—as he says—water. Therefore, the problem is related with the geometry or configuration of the barrel. If the barrel need not be opened, then by the common sense he mentions, the only possible action is to tilt the barrel to "see" what happens.

There lies the solution: we have to slowly tilt the barrel until the liquid level is in "correct" position. In his own words:

All that is necessary is to tilt the barrel as in Fig. 1, and if the edge of the surface of the water exactly touches the lip a at the same time that it touches the edge of the bottom b, it will be just half full. To be more exact, if the bottom is an inch or so from the ground, then we can allow for that, and the thickness of the bottom, at the top. If when the surface of the water reached the lip a it had risen to the point c in Fig. 2, then it would be more than half full. If, as in Fig. 3, some portion of the bottom were visible and the level of the water fell to the point d, then it would be less than half full.

This method applies to all symmetrically constructed vessels.

44 Selected Puzzles and Pastimes.

Download this EBook; all the problems are illustrated and perhaps you can make your own modifications.